> 2021 Answer = Easy Way using GMail (5 Mins)
We had a similar challenge to solve yesterday, and we solved it using a Google Apps Script!
Send Email From an HTML Form Without a Backend (Server) via Google!
The solution takes 5 mins to implement and I’ve documented with step-by-step instructions: https://github.com/nelsonic/html-form-send-email-via-google-script-without-server
Brief Overview
A. Using the sample script, deploy a Google App Script
Deploy the sample script as a Google Spreadsheet APP Script:
google-script-just-email.js

remember to set the
TO_ADDRESSin the script to where ever you want the emails to be sent.
and copy the APP URL so you can use it in the next step when you publish the script.
B. Create your HTML Form and Set the action to the App URL
Using the sample html file:
index.html
create a basic form.

remember to paste your APP URL into the form
actionin the HTML form.
C. Test the HTML Form in your Browser
Open the HTML Form in your Browser, Input some data & submit it!

Submit the form. You should see a confirmation that it was sent:

Open the inbox for the email address you set (above)

Done.
Everything about this is customisable, you can easily
style/theme the form with your favourite CSS Library
and Store the submitted data in a Google Spreadsheet
for quick analysis.
The complete instructions are available on GitHub:
https://github.com/nelsonic/html-form-send-email-via-google-script-without-server